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Question
In triangle ABC, prove the following:
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Solution
Let
Then,
Consider the LHS of the equation
\[LHS = a^2 \left( \cos^2 B - \cos^2 C \right) + b^2 \left( \cos^2 C - \cos^2 A \right) + c^2 \left( \cos^2 A - \cos^2 B \right)\]
\[ = k^2 \sin^2 A\left( 1 - \sin^2 B - 1 + \sin^2 C \right) + k^2 \sin^2 B\left( 1 - \sin^2 C - 1 + \sin^2 A \right) + k^2 \sin^2 C\left( 1 - \sin^2 A - 1 + \sin^2 B \right) \]
\[ = k^2 \sin^2 A\left( \sin^2 C - \sin^2 B \right) + k^2 \sin^2 B\left( \sin^2 A - \sin^2 C \right) + k^2 \sin^2 C\left( \sin^2 B - \sin^2 A \right)\]
\[ = k^2 \left( \sin^2 A \sin^2 C - \sin^2 A \sin^2 B + \sin^2 A \sin^2 B - \sin^2 B \sin^2 C + \sin^2 C \sin^2 B - \sin^2 C \sin^2 A \right)\]
\[ = k^2 \times 0 = 0 = RHS\]
\[\text{ Hence proved } .\]
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